LMFDB - Embedded newform 4225.2.a.ca.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 $ x^18 - 26x^16 + 281x^14 - 1632x^12 + 5482x^10 - 10620x^8 + 11052x^6 - 5165x^4 + 760x^2 - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$2.15692$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.15692 q^{2} +1.75445 q^{3} +2.65230 q^{4} +3.78422 q^{6} -3.86123 q^{7} +1.40696 q^{8} +0.0781086 q^{9} +O(q^{10})$	\(q+2.15692 q^{2} +1.75445 q^{3} +2.65230 q^{4} +3.78422 q^{6} -3.86123 q^{7} +1.40696 q^{8} +0.0781086 q^{9} -4.59986 q^{11} +4.65334 q^{12} -8.32836 q^{14} -2.26990 q^{16} +0.909776 q^{17} +0.168474 q^{18} +4.48893 q^{19} -6.77435 q^{21} -9.92154 q^{22} -5.76296 q^{23} +2.46845 q^{24} -5.12632 q^{27} -10.2411 q^{28} -1.52926 q^{29} -4.50050 q^{31} -7.70991 q^{32} -8.07025 q^{33} +1.96231 q^{34} +0.207168 q^{36} +3.73128 q^{37} +9.68225 q^{38} -3.71363 q^{41} -14.6117 q^{42} +4.80809 q^{43} -12.2002 q^{44} -12.4302 q^{46} -4.37026 q^{47} -3.98243 q^{48} +7.90910 q^{49} +1.59616 q^{51} +2.58550 q^{53} -11.0571 q^{54} -5.43261 q^{56} +7.87561 q^{57} -3.29848 q^{58} -13.2710 q^{59} +13.8206 q^{61} -9.70721 q^{62} -0.301595 q^{63} -12.0899 q^{64} -17.4069 q^{66} -5.30644 q^{67} +2.41300 q^{68} -10.1108 q^{69} -8.70613 q^{71} +0.109896 q^{72} +11.4102 q^{73} +8.04806 q^{74} +11.9060 q^{76} +17.7611 q^{77} +6.95233 q^{79} -9.22822 q^{81} -8.01000 q^{82} +1.81605 q^{83} -17.9676 q^{84} +10.3707 q^{86} -2.68301 q^{87} -6.47184 q^{88} -10.6440 q^{89} -15.2851 q^{92} -7.89591 q^{93} -9.42629 q^{94} -13.5267 q^{96} +2.34480 q^{97} +17.0593 q^{98} -0.359289 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9}+O(q^{10}) $ 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9	$ 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49} - 40 q^{51} - 90 q^{54} - 20 q^{56} - 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} - 72 q^{71} + 30 q^{74} - 4 q^{76} + 16 q^{79} - 30 q^{81} - 78 q^{84} + 30 q^{86} - 94 q^{89} - 128 q^{94} + 18 q^{96} - 76 q^{99}+O(q^{100}) $ 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9 - 22 * q^11 + 4 * q^14 - 12 * q^16 - 28 * q^19 - 26 * q^21 - 34 * q^24 - 20 * q^29 - 32 * q^31 - 18 * q^34 + 32 * q^36 - 52 * q^41 - 50 * q^44 - 30 * q^46 + 44 * q^49 - 40 * q^51 - 90 * q^54 - 20 * q^56 - 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 - 72 * q^71 + 30 * q^74 - 4 * q^76 + 16 * q^79 - 30 * q^81 - 78 * q^84 + 30 * q^86 - 94 * q^89 - 128 * q^94 + 18 * q^96 - 76 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.15692	1.52517	0.762586	−	0.646887i	$-0.223929\pi$
$2$	2.15692	1.52517	0.762586	+	0.646887i	$0.223929\pi$
$3$	1.75445	1.01293	0.506467	−	0.862259i	$-0.330951\pi$
$3$	1.75445	1.01293	0.506467	+	0.862259i	$0.330951\pi$
$4$	2.65230	1.32615
$5$	0	0
$5$	0	0
$6$	3.78422	1.54490
$7$	−3.86123	−1.45941	−0.729704	−	0.683763i	$-0.760342\pi$
$7$	−3.86123	−1.45941	−0.729704	+	0.683763i	$0.760342\pi$
$8$	1.40696	0.497436
$9$	0.0781086	0.0260362
$10$	0	0
$11$	−4.59986	−1.38691	−0.693456	−	0.720499i	$-0.743913\pi$
$11$	−4.59986	−1.38691	−0.693456	+	0.720499i	$0.743913\pi$
$12$	4.65334	1.34330
$13$	0	0
$13$	0	0
$14$	−8.32836	−2.22585
$15$	0	0
$16$	−2.26990	−0.567475
$17$	0.909776	0.220653	0.110327	−	0.993895i	$-0.464810\pi$
$17$	0.909776	0.220653	0.110327	+	0.993895i	$0.464810\pi$
$18$	0.168474	0.0397097
$19$	4.48893	1.02983	0.514915	−	0.857241i	$-0.327823\pi$
$19$	4.48893	1.02983	0.514915	+	0.857241i	$0.327823\pi$
$20$	0	0
$21$	−6.77435	−1.47828
$22$	−9.92154	−2.11528
$23$	−5.76296	−1.20166	−0.600830	−	0.799377i	$-0.705163\pi$
$23$	−5.76296	−1.20166	−0.600830	+	0.799377i	$0.705163\pi$
$24$	2.46845	0.503871
$25$	0	0
$26$	0	0
$27$	−5.12632	−0.986561
$28$	−10.2411	−1.93539
$29$	−1.52926	−0.283976	−0.141988	−	0.989868i	$-0.545349\pi$
$29$	−1.52926	−0.283976	−0.141988	+	0.989868i	$0.545349\pi$
$30$	0	0
$31$	−4.50050	−0.808313	−0.404156	−	0.914690i	$-0.632435\pi$
$31$	−4.50050	−0.808313	−0.404156	+	0.914690i	$0.632435\pi$
$32$	−7.70991	−1.36293
$33$	−8.07025	−1.40485
$34$	1.96231	0.336534
$35$	0	0
$36$	0.207168	0.0345279
$37$	3.73128	0.613418	0.306709	−	0.951803i	$-0.400772\pi$
$37$	3.73128	0.613418	0.306709	+	0.951803i	$0.400772\pi$
$38$	9.68225	1.57067
$39$	0	0
$40$	0	0
$41$	−3.71363	−0.579971	−0.289986	−	0.957031i	$-0.593651\pi$
$41$	−3.71363	−0.579971	−0.289986	+	0.957031i	$0.593651\pi$
$42$	−14.6117	−2.25464
$43$	4.80809	0.733226	0.366613	−	0.930373i	$-0.380517\pi$
$43$	4.80809	0.733226	0.366613	+	0.930373i	$0.380517\pi$
$44$	−12.2002	−1.83925
$45$	0	0
$46$	−12.4302	−1.83274
$47$	−4.37026	−0.637468	−0.318734	−	0.947844i	$-0.603258\pi$
$47$	−4.37026	−0.637468	−0.318734	+	0.947844i	$0.603258\pi$
$48$	−3.98243	−0.574815
$49$	7.90910	1.12987
$50$	0	0
$51$	1.59616	0.223507
$52$	0	0
$53$	2.58550	0.355146	0.177573	−	0.984108i	$-0.443175\pi$
$53$	2.58550	0.355146	0.177573	+	0.984108i	$0.443175\pi$
$54$	−11.0571	−1.50468
$55$	0	0
$56$	−5.43261	−0.725963
$57$	7.87561	1.04315
$58$	−3.29848	−0.433112
$59$	−13.2710	−1.72774	−0.863872	−	0.503712i	$-0.831967\pi$
$59$	−13.2710	−1.72774	−0.863872	+	0.503712i	$0.831967\pi$
$60$	0	0
$61$	13.8206	1.76955	0.884776	−	0.466017i	$-0.154311\pi$
$61$	13.8206	1.76955	0.884776	+	0.466017i	$0.154311\pi$
$62$	−9.70721	−1.23282
$63$	−0.301595	−0.0379974
$64$	−12.0899	−1.51123
$65$	0	0
$66$	−17.4069	−2.14264
$67$	−5.30644	−0.648285	−0.324142	−	0.946008i	$-0.605076\pi$
$67$	−5.30644	−0.648285	−0.324142	+	0.946008i	$0.605076\pi$
$68$	2.41300	0.292619
$69$	−10.1108	−1.21720
$70$	0	0
$71$	−8.70613	−1.03323	−0.516614	−	0.856219i	$-0.672808\pi$
$71$	−8.70613	−1.03323	−0.516614	+	0.856219i	$0.672808\pi$
$72$	0.109896	0.0129514
$73$	11.4102	1.33546	0.667729	−	0.744405i	$-0.267267\pi$
$73$	11.4102	1.33546	0.667729	+	0.744405i	$0.267267\pi$
$74$	8.04806	0.935568
$75$	0	0
$76$	11.9060	1.36571
$77$	17.7611	2.02407
$78$	0	0
$79$	6.95233	0.782199	0.391099	−	0.920349i	$-0.372095\pi$
$79$	6.95233	0.782199	0.391099	+	0.920349i	$0.372095\pi$
$80$	0	0
$81$	−9.22822	−1.02536
$82$	−8.01000	−0.884557
$83$	1.81605	0.199337	0.0996687	−	0.995021i	$-0.468222\pi$
$83$	1.81605	0.199337	0.0996687	+	0.995021i	$0.468222\pi$
$84$	−17.9676	−1.96043
$85$	0	0
$86$	10.3707	1.11830
$87$	−2.68301	−0.287649
$88$	−6.47184	−0.689900
$89$	−10.6440	−1.12826	−0.564132	−	0.825684i	$-0.690789\pi$
$89$	−10.6440	−1.12826	−0.564132	+	0.825684i	$0.690789\pi$
$90$	0	0
$91$	0	0
$92$	−15.2851	−1.59358
$93$	−7.89591	−0.818768
$94$	−9.42629	−0.972248
$95$	0	0
$96$	−13.5267	−1.38056
$97$	2.34480	0.238078	0.119039	−	0.992890i	$-0.462019\pi$
$97$	2.34480	0.238078	0.119039	+	0.992890i	$0.462019\pi$
$98$	17.0593	1.72325
$99$	−0.359289	−0.0361099
$100$	0	0
$101$	1.76997	0.176119	0.0880593	−	0.996115i	$-0.471934\pi$
$101$	1.76997	0.176119	0.0880593	+	0.996115i	$0.471934\pi$
$102$	3.44279	0.340887
$103$	7.02212	0.691910	0.345955	−	0.938251i	$-0.387555\pi$
$103$	7.02212	0.691910	0.345955	+	0.938251i	$0.387555\pi$
$104$	0	0
$105$	0	0
$106$	5.57671	0.541658
$107$	−8.92532	−0.862843	−0.431422	−	0.902150i	$-0.641988\pi$
$107$	−8.92532	−0.862843	−0.431422	+	0.902150i	$0.641988\pi$
$108$	−13.5966	−1.30833
$109$	11.9423	1.14386	0.571932	−	0.820301i	$-0.306194\pi$
$109$	11.9423	1.14386	0.571932	+	0.820301i	$0.306194\pi$
$110$	0	0
$111$	6.54635	0.621352
$112$	8.76460	0.828177
$113$	4.39725	0.413659	0.206829	−	0.978377i	$-0.433686\pi$
$113$	4.39725	0.413659	0.206829	+	0.978377i	$0.433686\pi$
$114$	16.9871	1.59098
$115$	0	0
$116$	−4.05605	−0.376595
$117$	0	0
$118$	−28.6246	−2.63511
$119$	−3.51285	−0.322023
$120$	0	0
$121$	10.1587	0.923522
$122$	29.8100	2.69887
$123$	−6.51539	−0.587473
$124$	−11.9367	−1.07194
$125$	0	0
$126$	−0.650517	−0.0579526
$127$	−12.2701	−1.08880	−0.544399	−	0.838827i	$-0.683242\pi$
$127$	−12.2701	−1.08880	−0.544399	+	0.838827i	$0.683242\pi$
$128$	−10.6570	−0.941958
$129$	8.43557	0.742710
$130$	0	0
$131$	11.4048	0.996440	0.498220	−	0.867051i	$-0.333987\pi$
$131$	11.4048	0.996440	0.498220	+	0.867051i	$0.333987\pi$
$132$	−21.4047	−1.86304
$133$	−17.3328	−1.50294
$134$	−11.4456	−0.988746
$135$	0	0
$136$	1.28002	0.109761
$137$	−7.32640	−0.625937	−0.312968	−	0.949764i	$-0.601323\pi$
$137$	−7.32640	−0.625937	−0.312968	+	0.949764i	$0.601323\pi$
$138$	−21.8083	−1.85644
$139$	8.30587	0.704495	0.352247	−	0.935907i	$-0.385418\pi$
$139$	8.30587	0.704495	0.352247	+	0.935907i	$0.385418\pi$
$140$	0	0
$141$	−7.66741	−0.645713
$142$	−18.7784	−1.57585
$143$	0	0
$144$	−0.177299	−0.0147749
$145$	0	0
$146$	24.6108	2.03680
$147$	13.8761	1.14449
$148$	9.89647	0.813485
$149$	−19.2533	−1.57729	−0.788647	−	0.614846i	$-0.789218\pi$
$149$	−19.2533	−1.57729	−0.788647	+	0.614846i	$0.789218\pi$
$150$	0	0
$151$	−2.40742	−0.195913	−0.0979564	−	0.995191i	$-0.531231\pi$
$151$	−2.40742	−0.195913	−0.0979564	+	0.995191i	$0.531231\pi$
$152$	6.31575	0.512275
$153$	0.0710613	0.00574497
$154$	38.3093	3.08705
$155$	0	0
$156$	0	0
$157$	−9.43852	−0.753276	−0.376638	−	0.926360i	$-0.622920\pi$
$157$	−9.43852	−0.753276	−0.376638	+	0.926360i	$0.622920\pi$
$158$	14.9956	1.19299
$159$	4.53614	0.359739
$160$	0	0
$161$	22.2521	1.75371
$162$	−19.9045	−1.56385
$163$	−19.2950	−1.51130	−0.755652	−	0.654974i	$-0.772680\pi$
$163$	−19.2950	−1.51130	−0.755652	+	0.654974i	$0.772680\pi$
$164$	−9.84967	−0.769130
$165$	0	0
$166$	3.91708	0.304024
$167$	−2.02546	−0.156734	−0.0783672	−	0.996925i	$-0.524971\pi$
$167$	−2.02546	−0.156734	−0.0783672	+	0.996925i	$0.524971\pi$
$168$	−9.53126	−0.735353
$169$	0	0
$170$	0	0
$171$	0.350624	0.0268129
$172$	12.7525	0.972369
$173$	7.66023	0.582396	0.291198	−	0.956663i	$-0.405946\pi$
$173$	7.66023	0.582396	0.291198	+	0.956663i	$0.405946\pi$
$174$	−5.78704	−0.438714
$175$	0	0
$176$	10.4412	0.787037
$177$	−23.2834	−1.75009
$178$	−22.9583	−1.72080
$179$	0.357735	0.0267384	0.0133692	−	0.999911i	$-0.495744\pi$
$179$	0.357735	0.0267384	0.0133692	+	0.999911i	$0.495744\pi$
$180$	0	0
$181$	2.73777	0.203497	0.101748	−	0.994810i	$-0.467556\pi$
$181$	2.73777	0.203497	0.101748	+	0.994810i	$0.467556\pi$
$182$	0	0
$183$	24.2477	1.79244
$184$	−8.10827	−0.597749
$185$	0	0
$186$	−17.0308	−1.24876
$187$	−4.18485	−0.306026
$188$	−11.5912	−0.845378
$189$	19.7939	1.43980
$190$	0	0
$191$	−22.2946	−1.61318	−0.806589	−	0.591112i	$-0.798689\pi$
$191$	−22.2946	−1.61318	−0.806589	+	0.591112i	$0.798689\pi$
$192$	−21.2111	−1.53078
$193$	8.27584	0.595708	0.297854	−	0.954611i	$-0.403729\pi$
$193$	8.27584	0.595708	0.297854	+	0.954611i	$0.403729\pi$
$194$	5.05754	0.363110
$195$	0	0
$196$	20.9773	1.49838
$197$	−8.26783	−0.589058	−0.294529	−	0.955642i	$-0.595163\pi$
$197$	−8.26783	−0.589058	−0.294529	+	0.955642i	$0.595163\pi$
$198$	−0.774957	−0.0550738
$199$	7.49085	0.531013	0.265506	−	0.964109i	$-0.414461\pi$
$199$	7.49085	0.531013	0.265506	+	0.964109i	$0.414461\pi$
$200$	0	0
$201$	−9.30991	−0.656670
$202$	3.81768	0.268611
$203$	5.90481	0.414437
$204$	4.23350	0.296404
$205$	0	0
$206$	15.1461	1.05528
$207$	−0.450136	−0.0312866
$208$	0	0
$209$	−20.6484	−1.42828
$210$	0	0
$211$	−4.48872	−0.309016	−0.154508	−	0.987992i	$-0.549379\pi$
$211$	−4.48872	−0.309016	−0.154508	+	0.987992i	$0.549379\pi$
$212$	6.85752	0.470977
$213$	−15.2745	−1.04659
$214$	−19.2512	−1.31599
$215$	0	0
$216$	−7.21255	−0.490752
$217$	17.3774	1.17966
$218$	25.7586	1.74459
$219$	20.0186	1.35273
$220$	0	0
$221$	0	0
$222$	14.1200	0.947669
$223$	23.3693	1.56493	0.782464	−	0.622696i	$-0.213963\pi$
$223$	23.3693	1.56493	0.782464	+	0.622696i	$0.213963\pi$
$224$	29.7697	1.98908
$225$	0	0
$226$	9.48452	0.630901
$227$	16.8003	1.11508	0.557538	−	0.830152i	$-0.311746\pi$
$227$	16.8003	1.11508	0.557538	+	0.830152i	$0.311746\pi$
$228$	20.8885	1.38337
$229$	14.9997	0.991206	0.495603	−	0.868549i	$-0.334947\pi$
$229$	14.9997	0.991206	0.495603	+	0.868549i	$0.334947\pi$
$230$	0	0
$231$	31.1611	2.05025
$232$	−2.15161	−0.141260
$233$	3.62537	0.237506	0.118753	−	0.992924i	$-0.462110\pi$
$233$	3.62537	0.237506	0.118753	+	0.992924i	$0.462110\pi$
$234$	0	0
$235$	0	0
$236$	−35.1988	−2.29125
$237$	12.1975	0.792316
$238$	−7.57694	−0.491140
$239$	−16.2842	−1.05333	−0.526667	−	0.850071i	$-0.676559\pi$
$239$	−16.2842	−1.05333	−0.526667	+	0.850071i	$0.676559\pi$
$240$	0	0
$241$	−13.6098	−0.876683	−0.438341	−	0.898809i	$-0.644434\pi$
$241$	−13.6098	−0.876683	−0.438341	+	0.898809i	$0.644434\pi$
$242$	21.9116	1.40853
$243$	−0.811524	−0.0520593
$244$	36.6565	2.34669
$245$	0	0
$246$	−14.0532	−0.895998
$247$	0	0
$248$	−6.33203	−0.402084
$249$	3.18618	0.201916
$250$	0	0
$251$	−1.86531	−0.117737	−0.0588685	−	0.998266i	$-0.518749\pi$
$251$	−1.86531	−0.117737	−0.0588685	+	0.998266i	$0.518749\pi$
$252$	−0.799922	−0.0503903
$253$	26.5088	1.66659
$254$	−26.4657	−1.66060
$255$	0	0
$256$	1.19335	0.0745843
$257$	22.0829	1.37749	0.688747	−	0.725002i	$-0.258161\pi$
$257$	22.0829	1.37749	0.688747	+	0.725002i	$0.258161\pi$
$258$	18.1948	1.13276
$259$	−14.4073	−0.895227
$260$	0	0
$261$	−0.119448	−0.00739365
$262$	24.5992	1.51974
$263$	−1.48996	−0.0918750	−0.0459375	−	0.998944i	$-0.514628\pi$
$263$	−1.48996	−0.0918750	−0.0459375	+	0.998944i	$0.514628\pi$
$264$	−11.3545	−0.698824
$265$	0	0
$266$	−37.3854	−2.29225
$267$	−18.6745	−1.14286
$268$	−14.0743	−0.859724
$269$	2.25757	0.137647	0.0688233	−	0.997629i	$-0.478076\pi$
$269$	2.25757	0.137647	0.0688233	+	0.997629i	$0.478076\pi$
$270$	0	0
$271$	−0.995573	−0.0604767	−0.0302384	−	0.999543i	$-0.509627\pi$
$271$	−0.995573	−0.0604767	−0.0302384	+	0.999543i	$0.509627\pi$
$272$	−2.06510	−0.125215
$273$	0	0
$274$	−15.8025	−0.954662
$275$	0	0
$276$	−26.8170	−1.61419
$277$	0.961303	0.0577591	0.0288795	−	0.999583i	$-0.490806\pi$
$277$	0.961303	0.0577591	0.0288795	+	0.999583i	$0.490806\pi$
$278$	17.9151	1.07448
$279$	−0.351527	−0.0210454
$280$	0	0
$281$	26.5703	1.58505	0.792526	−	0.609838i	$-0.208766\pi$
$281$	26.5703	1.58505	0.792526	+	0.609838i	$0.208766\pi$
$282$	−16.5380	−0.984823
$283$	−13.5742	−0.806899	−0.403450	−	0.915002i	$-0.632189\pi$
$283$	−13.5742	−0.806899	−0.403450	+	0.915002i	$0.632189\pi$
$284$	−23.0913	−1.37022
$285$	0	0
$286$	0	0
$287$	14.3392	0.846415
$288$	−0.602210	−0.0354856
$289$	−16.1723	−0.951312
$290$	0	0
$291$	4.11384	0.241157
$292$	30.2632	1.77102
$293$	−20.2641	−1.18384	−0.591919	−	0.805997i	$-0.701630\pi$
$293$	−20.2641	−1.18384	−0.591919	+	0.805997i	$0.701630\pi$
$294$	29.9297	1.74554
$295$	0	0
$296$	5.24977	0.305137
$297$	23.5804	1.36827
$298$	−41.5279	−2.40565
$299$	0	0
$300$	0	0
$301$	−18.5651	−1.07008
$302$	−5.19260	−0.298801
$303$	3.10533	0.178397
$304$	−10.1894	−0.584402
$305$	0	0
$306$	0.153274	0.00876206
$307$	21.8607	1.24765	0.623827	−	0.781563i	$-0.285577\pi$
$307$	21.8607	1.24765	0.623827	+	0.781563i	$0.285577\pi$
$308$	47.1079	2.68422
$309$	12.3200	0.700859
$310$	0	0
$311$	−9.22086	−0.522867	−0.261434	−	0.965221i	$-0.584195\pi$
$311$	−9.22086	−0.522867	−0.261434	+	0.965221i	$0.584195\pi$
$312$	0	0
$313$	−11.0528	−0.624741	−0.312371	−	0.949960i	$-0.601123\pi$
$313$	−11.0528	−0.624741	−0.312371	+	0.949960i	$0.601123\pi$
$314$	−20.3581	−1.14888
$315$	0	0
$316$	18.4397	1.03731
$317$	−22.4487	−1.26084	−0.630422	−	0.776252i	$-0.717118\pi$
$317$	−22.4487	−1.26084	−0.630422	+	0.776252i	$0.717118\pi$
$318$	9.78409	0.548664
$319$	7.03437	0.393849
$320$	0	0
$321$	−15.6591	−0.874004
$322$	47.9960	2.67471
$323$	4.08392	0.227235
$324$	−24.4760	−1.35978
$325$	0	0
$326$	−41.6178	−2.30500
$327$	20.9522	1.15866
$328$	−5.22494	−0.288499
$329$	16.8746	0.930325
$330$	0	0
$331$	28.4167	1.56192	0.780961	−	0.624580i	$-0.214730\pi$
$331$	28.4167	1.56192	0.780961	+	0.624580i	$0.214730\pi$
$332$	4.81672	0.264352
$333$	0.291445	0.0159711
$334$	−4.36875	−0.239047
$335$	0	0
$336$	15.3771	0.838889
$337$	16.5065	0.899169	0.449584	−	0.893238i	$-0.351572\pi$
$337$	16.5065	0.899169	0.449584	+	0.893238i	$0.351572\pi$
$338$	0	0
$339$	7.71477	0.419009
$340$	0	0
$341$	20.7017	1.12106
$342$	0.756267	0.0408942
$343$	−3.51023	−0.189535
$344$	6.76480	0.364734
$345$	0	0
$346$	16.5225	0.888255
$347$	−0.250520	−0.0134486	−0.00672431	−	0.999977i	$-0.502140\pi$
$347$	−0.250520	−0.0134486	−0.00672431	+	0.999977i	$0.502140\pi$
$348$	−7.11615	−0.381466
$349$	−13.9617	−0.747354	−0.373677	−	0.927559i	$-0.621903\pi$
$349$	−13.9617	−0.747354	−0.373677	+	0.927559i	$0.621903\pi$
$350$	0	0
$351$	0	0
$352$	35.4646	1.89027
$353$	12.0435	0.641010	0.320505	−	0.947247i	$-0.396147\pi$
$353$	12.0435	0.641010	0.320505	+	0.947247i	$0.396147\pi$
$354$	−50.2205	−2.66919
$355$	0	0
$356$	−28.2312	−1.49625
$357$	−6.16314	−0.326188
$358$	0.771606	0.0407806
$359$	−16.0890	−0.849145	−0.424573	−	0.905394i	$-0.639576\pi$
$359$	−16.0890	−0.849145	−0.424573	+	0.905394i	$0.639576\pi$
$360$	0	0
$361$	1.15045	0.0605500
$362$	5.90515	0.310368
$363$	17.8230	0.935467
$364$	0	0
$365$	0	0
$366$	52.3003	2.73378
$367$	−10.7292	−0.560062	−0.280031	−	0.959991i	$-0.590345\pi$
$367$	−10.7292	−0.560062	−0.280031	+	0.959991i	$0.590345\pi$
$368$	13.0813	0.681911
$369$	−0.290066	−0.0151002
$370$	0	0
$371$	−9.98320	−0.518302
$372$	−20.9423	−1.08581
$373$	−30.2882	−1.56826	−0.784132	−	0.620595i	$-0.786891\pi$
$373$	−30.2882	−1.56826	−0.784132	+	0.620595i	$0.786891\pi$
$374$	−9.02637	−0.466743
$375$	0	0
$376$	−6.14879	−0.317100
$377$	0	0
$378$	42.6939	2.19594
$379$	−30.1599	−1.54921	−0.774605	−	0.632445i	$-0.782051\pi$
$379$	−30.1599	−1.54921	−0.774605	+	0.632445i	$0.782051\pi$
$380$	0	0
$381$	−21.5274	−1.10288
$382$	−48.0876	−2.46038
$383$	−12.5699	−0.642291	−0.321145	−	0.947030i	$-0.604068\pi$
$383$	−12.5699	−0.642291	−0.321145	+	0.947030i	$0.604068\pi$
$384$	−18.6973	−0.954142
$385$	0	0
$386$	17.8503	0.908558
$387$	0.375553	0.0190904
$388$	6.21911	0.315727
$389$	−31.6494	−1.60469	−0.802345	−	0.596860i	$-0.796415\pi$
$389$	−31.6494	−1.60469	−0.802345	+	0.596860i	$0.796415\pi$
$390$	0	0
$391$	−5.24300	−0.265150
$392$	11.1278	0.562039
$393$	20.0092	1.00933
$394$	−17.8330	−0.898416
$395$	0	0
$396$	−0.952942	−0.0478872
$397$	12.1544	0.610012	0.305006	−	0.952350i	$-0.401341\pi$
$397$	12.1544	0.610012	0.305006	+	0.952350i	$0.401341\pi$
$398$	16.1572	0.809886
$399$	−30.4096	−1.52238
$400$	0	0
$401$	−22.3511	−1.11616	−0.558079	−	0.829788i	$-0.688461\pi$
$401$	−22.3511	−1.11616	−0.558079	+	0.829788i	$0.688461\pi$
$402$	−20.0807	−1.00154
$403$	0	0
$404$	4.69449	0.233560
$405$	0	0
$406$	12.7362	0.632087
$407$	−17.1634	−0.850756
$408$	2.24574	0.111181
$409$	28.0318	1.38608	0.693042	−	0.720897i	$-0.256270\pi$
$409$	28.0318	1.38608	0.693042	+	0.720897i	$0.256270\pi$
$410$	0	0
$411$	−12.8538	−0.634033
$412$	18.6248	0.917577
$413$	51.2426	2.52148
$414$	−0.970908	−0.0477175
$415$	0	0
$416$	0	0
$417$	14.5723	0.713607
$418$	−44.5370	−2.17838
$419$	1.49434	0.0730035	0.0365018	−	0.999334i	$-0.488379\pi$
$419$	1.49434	0.0730035	0.0365018	+	0.999334i	$0.488379\pi$
$420$	0	0
$421$	−13.5682	−0.661274	−0.330637	−	0.943758i	$-0.607264\pi$
$421$	−13.5682	−0.661274	−0.330637	+	0.943758i	$0.607264\pi$
$422$	−9.68180	−0.471303
$423$	−0.341355	−0.0165972
$424$	3.63770	0.176662
$425$	0	0
$426$	−32.9459	−1.59623
$427$	−53.3647	−2.58250
$428$	−23.6726	−1.14426
$429$	0	0
$430$	0	0
$431$	4.50300	0.216902	0.108451	−	0.994102i	$-0.465411\pi$
$431$	4.50300	0.216902	0.108451	+	0.994102i	$0.465411\pi$
$432$	11.6362	0.559849
$433$	27.9595	1.34365	0.671825	−	0.740710i	$-0.265511\pi$
$433$	27.9595	1.34365	0.671825	+	0.740710i	$0.265511\pi$
$434$	37.4818	1.79918
$435$	0	0
$436$	31.6746	1.51694
$437$	−25.8695	−1.23751
$438$	43.1785	2.06315
$439$	−31.0146	−1.48025	−0.740124	−	0.672470i	$-0.765233\pi$
$439$	−31.0146	−1.48025	−0.740124	+	0.672470i	$0.765233\pi$
$440$	0	0
$441$	0.617768	0.0294175
$442$	0	0
$443$	37.1681	1.76591	0.882955	−	0.469457i	$-0.155550\pi$
$443$	37.1681	1.76591	0.882955	+	0.469457i	$0.155550\pi$
$444$	17.3629	0.824007
$445$	0	0
$446$	50.4058	2.38678
$447$	−33.7791	−1.59770
$448$	46.6818	2.20551
$449$	−39.2051	−1.85020	−0.925102	−	0.379718i	$-0.876021\pi$
$449$	−39.2051	−1.85020	−0.925102	+	0.379718i	$0.876021\pi$
$450$	0	0
$451$	17.0822	0.804369
$452$	11.6628	0.548574
$453$	−4.22370	−0.198447
$454$	36.2369	1.70068
$455$	0	0
$456$	11.0807	0.518901
$457$	27.9180	1.30595	0.652974	−	0.757380i	$-0.273521\pi$
$457$	27.9180	1.30595	0.652974	+	0.757380i	$0.273521\pi$
$458$	32.3531	1.51176
$459$	−4.66381	−0.217688
$460$	0	0
$461$	39.6855	1.84834	0.924170	−	0.381982i	$-0.124758\pi$
$461$	39.6855	1.84834	0.924170	+	0.381982i	$0.124758\pi$
$462$	67.2120	3.12698
$463$	5.21385	0.242308	0.121154	−	0.992634i	$-0.461340\pi$
$463$	5.21385	0.242308	0.121154	+	0.992634i	$0.461340\pi$
$464$	3.47126	0.161149
$465$	0	0
$466$	7.81963	0.362237
$467$	−29.5295	−1.36646	−0.683230	−	0.730203i	$-0.739425\pi$
$467$	−29.5295	−1.36646	−0.683230	+	0.730203i	$0.739425\pi$
$468$	0	0
$469$	20.4894	0.946112
$470$	0	0
$471$	−16.5595	−0.763019
$472$	−18.6719	−0.859442
$473$	−22.1165	−1.01692
$474$	26.3091	1.20842
$475$	0	0
$476$	−9.31715	−0.427051
$477$	0.201950	0.00924664
$478$	−35.1236	−1.60652
$479$	−8.16917	−0.373259	−0.186629	−	0.982430i	$-0.559756\pi$
$479$	−8.16917	−0.373259	−0.186629	+	0.982430i	$0.559756\pi$
$480$	0	0
$481$	0	0
$482$	−29.3552	−1.33709
$483$	39.0403	1.77639
$484$	26.9441	1.22473
$485$	0	0
$486$	−1.75039	−0.0793994
$487$	20.1234	0.911879	0.455940	−	0.890011i	$-0.349303\pi$
$487$	20.1234	0.911879	0.455940	+	0.890011i	$0.349303\pi$
$488$	19.4451	0.880240
$489$	−33.8522	−1.53085
$490$	0	0
$491$	−39.7304	−1.79301	−0.896504	−	0.443035i	$-0.853902\pi$
$491$	−39.7304	−1.79301	−0.896504	+	0.443035i	$0.853902\pi$
$492$	−17.2808	−0.779078
$493$	−1.39128	−0.0626601
$494$	0	0
$495$	0	0
$496$	10.2157	0.458697
$497$	33.6164	1.50790
$498$	6.87233	0.307956
$499$	−17.8068	−0.797143	−0.398571	−	0.917137i	$-0.630494\pi$
$499$	−17.8068	−0.797143	−0.398571	+	0.917137i	$0.630494\pi$
$500$	0	0
$501$	−3.55357	−0.158762
$502$	−4.02331	−0.179569
$503$	18.0669	0.805562	0.402781	−	0.915296i	$-0.368044\pi$
$503$	18.0669	0.805562	0.402781	+	0.915296i	$0.368044\pi$
$504$	−0.424333	−0.0189013
$505$	0	0
$506$	57.1774	2.54184
$507$	0	0
$508$	−32.5441	−1.44391
$509$	10.9295	0.484443	0.242221	−	0.970221i	$-0.422124\pi$
$509$	10.9295	0.484443	0.242221	+	0.970221i	$0.422124\pi$
$510$	0	0
$511$	−44.0572	−1.94898
$512$	23.8880	1.05571
$513$	−23.0117	−1.01599
$514$	47.6311	2.10092
$515$	0	0
$516$	22.3737	0.984946
$517$	20.1026	0.884111
$518$	−31.0754	−1.36538
$519$	13.4395	0.589929
$520$	0	0
$521$	26.2980	1.15213	0.576067	−	0.817402i	$-0.304587\pi$
$521$	26.2980	1.15213	0.576067	+	0.817402i	$0.304587\pi$
$522$	−0.257640	−0.0112766
$523$	−13.4897	−0.589862	−0.294931	−	0.955519i	$-0.595297\pi$
$523$	−13.4897	−0.589862	−0.294931	+	0.955519i	$0.595297\pi$
$524$	30.2489	1.32143
$525$	0	0
$526$	−3.21373	−0.140125
$527$	−4.09444	−0.178357
$528$	18.3186	0.797217
$529$	10.2117	0.443986
$530$	0	0
$531$	−1.03658	−0.0449839
$532$	−45.9717	−1.99313
$533$	0	0
$534$	−40.2793	−1.74306
$535$	0	0
$536$	−7.46597	−0.322481
$537$	0.627630	0.0270842
$538$	4.86940	0.209935
$539$	−36.3808	−1.56703
$540$	0	0
$541$	−45.7507	−1.96698	−0.983488	−	0.180973i	$-0.942075\pi$
$541$	−45.7507	−1.96698	−0.983488	+	0.180973i	$0.942075\pi$
$542$	−2.14737	−0.0922375
$543$	4.80329	0.206129
$544$	−7.01429	−0.300735
$545$	0	0
$546$	0	0
$547$	−9.40228	−0.402012	−0.201006	−	0.979590i	$-0.564421\pi$
$547$	−9.40228	−0.402012	−0.201006	+	0.979590i	$0.564421\pi$
$548$	−19.4318	−0.830087
$549$	1.07951	0.0460724
$550$	0	0
$551$	−6.86472	−0.292447
$552$	−14.2256	−0.605481
$553$	−26.8446	−1.14155
$554$	2.07345	0.0880926
$555$	0	0
$556$	22.0297	0.934266
$557$	−5.18137	−0.219542	−0.109771	−	0.993957i	$-0.535012\pi$
$557$	−5.18137	−0.219542	−0.109771	+	0.993957i	$0.535012\pi$
$558$	−0.758216	−0.0320979
$559$	0	0
$560$	0	0
$561$	−7.34212	−0.309984
$562$	57.3100	2.41748
$563$	−28.8360	−1.21529	−0.607646	−	0.794208i	$-0.707886\pi$
$563$	−28.8360	−1.21529	−0.607646	+	0.794208i	$0.707886\pi$
$564$	−20.3363	−0.856313
$565$	0	0
$566$	−29.2783	−1.23066
$567$	35.6323	1.49642
$568$	−12.2492	−0.513965
$569$	18.8502	0.790241	0.395121	−	0.918629i	$-0.370703\pi$
$569$	18.8502	0.790241	0.395121	+	0.918629i	$0.370703\pi$
$570$	0	0
$571$	−17.1912	−0.719430	−0.359715	−	0.933062i	$-0.617126\pi$
$571$	−17.1912	−0.719430	−0.359715	+	0.933062i	$0.617126\pi$
$572$	0	0
$573$	−39.1148	−1.63404
$574$	30.9285	1.29093
$575$	0	0
$576$	−0.944322	−0.0393468
$577$	−1.61025	−0.0670358	−0.0335179	−	0.999438i	$-0.510671\pi$
$577$	−1.61025	−0.0670358	−0.0335179	+	0.999438i	$0.510671\pi$
$578$	−34.8824	−1.45092
$579$	14.5196	0.603414
$580$	0	0
$581$	−7.01219	−0.290915
$582$	8.87322	0.367807
$583$	−11.8929	−0.492555
$584$	16.0537	0.664305
$585$	0	0
$586$	−43.7079	−1.80556
$587$	20.9373	0.864175	0.432088	−	0.901832i	$-0.357777\pi$
$587$	20.9373	0.864175	0.432088	+	0.901832i	$0.357777\pi$
$588$	36.8037	1.51776
$589$	−20.2024	−0.832425
$590$	0	0
$591$	−14.5055	−0.596678
$592$	−8.46962	−0.348099
$593$	−29.3903	−1.20692	−0.603458	−	0.797395i	$-0.706211\pi$
$593$	−29.3903	−1.20692	−0.603458	+	0.797395i	$0.706211\pi$
$594$	50.8610	2.08685
$595$	0	0
$596$	−51.0657	−2.09173
$597$	13.1424	0.537881
$598$	0	0
$599$	23.4111	0.956550	0.478275	−	0.878210i	$-0.341262\pi$
$599$	23.4111	0.956550	0.478275	+	0.878210i	$0.341262\pi$
$600$	0	0
$601$	−20.5700	−0.839069	−0.419535	−	0.907739i	$-0.637807\pi$
$601$	−20.5700	−0.839069	−0.419535	+	0.907739i	$0.637807\pi$
$602$	−40.0435	−1.63205
$603$	−0.414479	−0.0168789
$604$	−6.38519	−0.259810
$605$	0	0
$606$	6.69795	0.272086
$607$	−10.6454	−0.432082	−0.216041	−	0.976384i	$-0.569315\pi$
$607$	−10.6454	−0.432082	−0.216041	+	0.976384i	$0.569315\pi$
$608$	−34.6092	−1.40359
$609$	10.3597	0.419797
$610$	0	0
$611$	0	0
$612$	0.188476	0.00761869
$613$	−41.6219	−1.68109	−0.840546	−	0.541740i	$-0.817766\pi$
$613$	−41.6219	−1.68109	−0.840546	+	0.541740i	$0.817766\pi$
$614$	47.1517	1.90289
$615$	0	0
$616$	24.9892	1.00685
$617$	−23.6772	−0.953208	−0.476604	−	0.879118i	$-0.658132\pi$
$617$	−23.6772	−0.953208	−0.476604	+	0.879118i	$0.658132\pi$
$618$	26.5732	1.06893
$619$	−11.5271	−0.463314	−0.231657	−	0.972798i	$-0.574415\pi$
$619$	−11.5271	−0.463314	−0.231657	+	0.972798i	$0.574415\pi$
$620$	0	0
$621$	29.5428	1.18551
$622$	−19.8887	−0.797462
$623$	41.0990	1.64660
$624$	0	0
$625$	0	0
$626$	−23.8400	−0.952838
$627$	−36.2267	−1.44676
$628$	−25.0338	−0.998958
$629$	3.39463	0.135353
$630$	0	0
$631$	−44.6970	−1.77936	−0.889680	−	0.456584i	$-0.849073\pi$
$631$	−44.6970	−1.77936	−0.889680	+	0.456584i	$0.849073\pi$
$632$	9.78167	0.389094
$633$	−7.87525	−0.313013
$634$	−48.4200	−1.92301
$635$	0	0
$636$	12.0312	0.477068
$637$	0	0
$638$	15.1726	0.600688
$639$	−0.680023	−0.0269013
$640$	0	0
$641$	−30.0561	−1.18714	−0.593572	−	0.804781i	$-0.702283\pi$
$641$	−30.0561	−1.18714	−0.593572	+	0.804781i	$0.702283\pi$
$642$	−33.7753	−1.33301
$643$	41.2166	1.62543	0.812713	−	0.582664i	$-0.197990\pi$
$643$	41.2166	1.62543	0.812713	+	0.582664i	$0.197990\pi$
$644$	59.0193	2.32569
$645$	0	0
$646$	8.80868	0.346573
$647$	−17.4713	−0.686868	−0.343434	−	0.939177i	$-0.611590\pi$
$647$	−17.4713	−0.686868	−0.343434	+	0.939177i	$0.611590\pi$
$648$	−12.9838	−0.510051
$649$	61.0450	2.39623
$650$	0	0
$651$	30.4879	1.19492
$652$	−51.1763	−2.00422
$653$	−11.2814	−0.441476	−0.220738	−	0.975333i	$-0.570847\pi$
$653$	−11.2814	−0.441476	−0.220738	+	0.975333i	$0.570847\pi$
$654$	45.1922	1.76716
$655$	0	0
$656$	8.42956	0.329119
$657$	0.891231	0.0347702
$658$	36.3971	1.41891
$659$	−4.14015	−0.161277	−0.0806386	−	0.996743i	$-0.525696\pi$
$659$	−4.14015	−0.161277	−0.0806386	+	0.996743i	$0.525696\pi$
$660$	0	0
$661$	1.63795	0.0637087	0.0318544	−	0.999493i	$-0.489859\pi$
$661$	1.63795	0.0637087	0.0318544	+	0.999493i	$0.489859\pi$
$662$	61.2925	2.38220
$663$	0	0
$664$	2.55512	0.0991577
$665$	0	0
$666$	0.628623	0.0243586
$667$	8.81304	0.341242
$668$	−5.37212	−0.207854
$669$	41.0004	1.58517
$670$	0	0
$671$	−63.5731	−2.45421
$672$	52.2297	2.01480
$673$	49.1173	1.89333	0.946667	−	0.322215i	$-0.104427\pi$
$673$	49.1173	1.89333	0.946667	+	0.322215i	$0.104427\pi$
$674$	35.6033	1.37139
$675$	0	0
$676$	0	0
$677$	50.3258	1.93418	0.967089	−	0.254438i	$-0.0818905\pi$
$677$	50.3258	1.93418	0.967089	+	0.254438i	$0.0818905\pi$
$678$	16.6401	0.639061
$679$	−9.05380	−0.347453
$680$	0	0
$681$	29.4754	1.12950
$682$	44.6518	1.70981
$683$	43.6035	1.66844	0.834222	−	0.551429i	$-0.185917\pi$
$683$	43.6035	1.66844	0.834222	+	0.551429i	$0.185917\pi$
$684$	0.929960	0.0355579
$685$	0	0
$686$	−7.57128	−0.289073
$687$	26.3162	1.00403
$688$	−10.9139	−0.416087
$689$	0	0
$690$	0	0
$691$	−47.9556	−1.82432	−0.912159	−	0.409837i	$-0.865585\pi$
$691$	−47.9556	−1.82432	−0.912159	+	0.409837i	$0.865585\pi$
$692$	20.3172	0.772345
$693$	1.38730	0.0526990
$694$	−0.540352	−0.0205115
$695$	0	0
$696$	−3.77490	−0.143087
$697$	−3.37857	−0.127972
$698$	−30.1143	−1.13984
$699$	6.36054	0.240578
$700$	0	0
$701$	0.0141658	0.000535035 0	0.000267518	−	1.00000i	$-0.499915\pi$
$701$	0.0141658	0.000535035 0	0.000267518		1.00000i	$0.499915\pi$
$702$	0	0
$703$	16.7494	0.631716
$704$	55.6117	2.09595
$705$	0	0
$706$	25.9768	0.977651
$707$	−6.83426	−0.257029
$708$	−61.7547	−2.32088
$709$	−30.2820	−1.13727	−0.568633	−	0.822591i	$-0.692528\pi$
$709$	−30.2820	−1.13727	−0.568633	+	0.822591i	$0.692528\pi$
$710$	0	0
$711$	0.543037	0.0203655
$712$	−14.9757	−0.561240
$713$	25.9362	0.971317
$714$	−13.2934	−0.497493
$715$	0	0
$716$	0.948821	0.0354591
$717$	−28.5698	−1.06696
$718$	−34.7027	−1.29509
$719$	−26.4073	−0.984826	−0.492413	−	0.870362i	$-0.663885\pi$
$719$	−26.4073	−0.984826	−0.492413	+	0.870362i	$0.663885\pi$
$720$	0	0
$721$	−27.1140	−1.00978
$722$	2.48143	0.0923492
$723$	−23.8777	−0.888022
$724$	7.26139	0.269867
$725$	0	0
$726$	38.4429	1.42675
$727$	−32.1909	−1.19389	−0.596947	−	0.802280i	$-0.703620\pi$
$727$	−32.1909	−1.19389	−0.596947	+	0.802280i	$0.703620\pi$
$728$	0	0
$729$	26.2609	0.972626
$730$	0	0
$731$	4.37428	0.161789
$732$	64.3122	2.37705
$733$	−31.1866	−1.15190	−0.575952	−	0.817483i	$-0.695369\pi$
$733$	−31.1866	−1.15190	−0.575952	+	0.817483i	$0.695369\pi$
$734$	−23.1421	−0.854191
$735$	0	0
$736$	44.4319	1.63778
$737$	24.4089	0.899114
$738$	−0.625650	−0.0230305
$739$	−7.56155	−0.278156	−0.139078	−	0.990281i	$-0.544414\pi$
$739$	−7.56155	−0.278156	−0.139078	+	0.990281i	$0.544414\pi$
$740$	0	0
$741$	0	0
$742$	−21.5330	−0.790500
$743$	10.1941	0.373986	0.186993	−	0.982361i	$-0.440126\pi$
$743$	10.1941	0.373986	0.186993	+	0.982361i	$0.440126\pi$
$744$	−11.1093	−0.407285
$745$	0	0
$746$	−65.3292	−2.39187
$747$	0.141849	0.00518999
$748$	−11.0995	−0.405837
$749$	34.4627	1.25924
$750$	0	0
$751$	−28.8163	−1.05152	−0.525760	−	0.850633i	$-0.676219\pi$
$751$	−28.8163	−1.05152	−0.525760	+	0.850633i	$0.676219\pi$
$752$	9.92004	0.361747
$753$	−3.27259	−0.119260
$754$	0	0
$755$	0	0
$756$	52.4994	1.90939
$757$	−22.1593	−0.805394	−0.402697	−	0.915333i	$-0.631927\pi$
$757$	−22.1593	−0.805394	−0.402697	+	0.915333i	$0.631927\pi$
$758$	−65.0525	−2.36281
$759$	46.5085	1.68815
$760$	0	0
$761$	4.04239	0.146536	0.0732682	−	0.997312i	$-0.476657\pi$
$761$	4.04239	0.146536	0.0732682	+	0.997312i	$0.476657\pi$
$762$	−46.4328	−1.68208
$763$	−46.1120	−1.66937
$764$	−59.1319	−2.13932
$765$	0	0
$766$	−27.1122	−0.979604
$767$	0	0
$768$	2.09368	0.0755491
$769$	−0.874220	−0.0315252	−0.0157626	−	0.999876i	$-0.505018\pi$
$769$	−0.874220	−0.0315252	−0.0157626	+	0.999876i	$0.505018\pi$
$770$	0	0
$771$	38.7434	1.39531
$772$	21.9500	0.789999
$773$	24.6865	0.887911	0.443956	−	0.896049i	$-0.353575\pi$
$773$	24.6865	0.887911	0.443956	+	0.896049i	$0.353575\pi$
$774$	0.810037	0.0291162
$775$	0	0
$776$	3.29904	0.118429
$777$	−25.2770	−0.906806
$778$	−68.2653	−2.44743
$779$	−16.6702	−0.597272
$780$	0	0
$781$	40.0470	1.43299
$782$	−11.3087	−0.404399
$783$	7.83947	0.280160
$784$	−17.9528	−0.641173
$785$	0	0
$786$	43.1582	1.53940
$787$	16.4977	0.588081	0.294040	−	0.955793i	$-0.405000\pi$
$787$	16.4977	0.588081	0.294040	+	0.955793i	$0.405000\pi$
$788$	−21.9288	−0.781180
$789$	−2.61407	−0.0930634
$790$	0	0
$791$	−16.9788	−0.603696
$792$	−0.505506	−0.0179624
$793$	0	0
$794$	26.2161	0.930374
$795$	0	0
$796$	19.8680	0.704203
$797$	−12.3809	−0.438553	−0.219276	−	0.975663i	$-0.570370\pi$
$797$	−12.3809	−0.438553	−0.219276	+	0.975663i	$0.570370\pi$
$798$	−65.5910	−2.32189
$799$	−3.97595	−0.140659
$800$	0	0
$801$	−0.831390	−0.0293757
$802$	−48.2094	−1.70233
$803$	−52.4852	−1.85216
$804$	−24.6927	−0.870844
$805$	0	0
$806$	0	0
$807$	3.96081	0.139427
$808$	2.49028	0.0876078
$809$	28.6006	1.00554	0.502771	−	0.864419i	$-0.332314\pi$
$809$	28.6006	1.00554	0.502771	+	0.864419i	$0.332314\pi$
$810$	0	0
$811$	24.4980	0.860242	0.430121	−	0.902771i	$-0.358471\pi$
$811$	24.4980	0.860242	0.430121	+	0.902771i	$0.358471\pi$
$812$	15.6613	0.549605
$813$	−1.74669	−0.0612590
$814$	−37.0200	−1.29755
$815$	0	0
$816$	−3.62312	−0.126835
$817$	21.5831	0.755099
$818$	60.4624	2.11402
$819$	0	0
$820$	0	0
$821$	29.3972	1.02597	0.512984	−	0.858398i	$-0.328540\pi$
$821$	29.3972	1.02597	0.512984	+	0.858398i	$0.328540\pi$
$822$	−27.7247	−0.967010
$823$	43.6578	1.52181	0.760907	−	0.648861i	$-0.224754\pi$
$823$	43.6578	1.52181	0.760907	+	0.648861i	$0.224754\pi$
$824$	9.87986	0.344181
$825$	0	0
$826$	110.526	3.84569
$827$	50.6049	1.75971	0.879853	−	0.475247i	$-0.157641\pi$
$827$	50.6049	1.75971	0.879853	+	0.475247i	$0.157641\pi$
$828$	−1.19390	−0.0414908
$829$	−25.3169	−0.879293	−0.439646	−	0.898171i	$-0.644896\pi$
$829$	−25.3169	−0.879293	−0.439646	+	0.898171i	$0.644896\pi$
$830$	0	0
$831$	1.68656	0.0585062
$832$	0	0
$833$	7.19551	0.249309
$834$	31.4312	1.08837
$835$	0	0
$836$	−54.7659	−1.89412
$837$	23.0710	0.797450
$838$	3.22318	0.111343
$839$	21.2608	0.734005	0.367002	−	0.930220i	$-0.380384\pi$
$839$	21.2608	0.734005	0.367002	+	0.930220i	$0.380384\pi$
$840$	0	0
$841$	−26.6614	−0.919358
$842$	−29.2656	−1.00856
$843$	46.6164	1.60555
$844$	−11.9054	−0.409802
$845$	0	0
$846$	−0.736274	−0.0253136
$847$	−39.2252	−1.34780
$848$	−5.86882	−0.201536
$849$	−23.8152	−0.817336
$850$	0	0
$851$	−21.5032	−0.737120
$852$	−40.5126	−1.38794
$853$	38.4505	1.31652	0.658261	−	0.752790i	$-0.271292\pi$
$853$	38.4505	1.31652	0.658261	+	0.752790i	$0.271292\pi$
$854$	−115.103	−3.93875
$855$	0	0
$856$	−12.5576	−0.429210
$857$	10.2334	0.349567	0.174783	−	0.984607i	$-0.444077\pi$
$857$	10.2334	0.349567	0.174783	+	0.984607i	$0.444077\pi$
$858$	0	0
$859$	4.85242	0.165562	0.0827812	−	0.996568i	$-0.473620\pi$
$859$	4.85242	0.165562	0.0827812	+	0.996568i	$0.473620\pi$
$860$	0	0
$861$	25.1574	0.857363
$862$	9.71262	0.330813
$863$	29.4296	1.00180	0.500898	−	0.865506i	$-0.333003\pi$
$863$	29.4296	1.00180	0.500898	+	0.865506i	$0.333003\pi$
$864$	39.5235	1.34462
$865$	0	0
$866$	60.3065	2.04930
$867$	−28.3736	−0.963617
$868$	46.0902	1.56440
$869$	−31.9798	−1.08484
$870$	0	0
$871$	0	0
$872$	16.8024	0.569000
$873$	0.183149	0.00619864
$874$	−55.7984	−1.88741
$875$	0	0
$876$	53.0954	1.79393
$877$	−27.8045	−0.938890	−0.469445	−	0.882962i	$-0.655546\pi$
$877$	−27.8045	−0.938890	−0.469445	+	0.882962i	$0.655546\pi$
$878$	−66.8961	−2.25763
$879$	−35.5523	−1.19915
$880$	0	0
$881$	−26.7991	−0.902883	−0.451442	−	0.892301i	$-0.649090\pi$
$881$	−26.7991	−0.902883	−0.451442	+	0.892301i	$0.649090\pi$
$882$	1.33248	0.0448668
$883$	28.7872	0.968765	0.484382	−	0.874856i	$-0.339044\pi$
$883$	28.7872	0.968765	0.484382	+	0.874856i	$0.339044\pi$
$884$	0	0
$885$	0	0
$886$	80.1686	2.69332
$887$	13.6374	0.457900	0.228950	−	0.973438i	$-0.426471\pi$
$887$	13.6374	0.457900	0.228950	+	0.973438i	$0.426471\pi$
$888$	9.21048	0.309083
$889$	47.3778	1.58900
$890$	0	0
$891$	42.4486	1.42208
$892$	61.9826	2.07533
$893$	−19.6178	−0.656483
$894$	−72.8588	−2.43676
$895$	0	0
$896$	41.1493	1.37470
$897$	0	0
$898$	−84.5623	−2.82188
$899$	6.88241	0.229541
$900$	0	0
$901$	2.35222	0.0783640
$902$	36.8449	1.22680
$903$	−32.5717	−1.08392
$904$	6.18677	0.205769
$905$	0	0
$906$	−9.11018	−0.302666
$907$	−1.96733	−0.0653241	−0.0326621	−	0.999466i	$-0.510399\pi$
$907$	−1.96733	−0.0653241	−0.0326621	+	0.999466i	$0.510399\pi$
$908$	44.5595	1.47876
$909$	0.138250	0.00458546
$910$	0	0
$911$	12.7181	0.421370	0.210685	−	0.977554i	$-0.432431\pi$
$911$	12.7181	0.421370	0.210685	+	0.977554i	$0.432431\pi$
$912$	−17.8768	−0.591961
$913$	−8.35359	−0.276463
$914$	60.2169	1.99180
$915$	0	0
$916$	39.7837	1.31449
$917$	−44.0365	−1.45421
$918$	−10.0595	−0.332011
$919$	2.60894	0.0860610	0.0430305	−	0.999074i	$-0.486299\pi$
$919$	2.60894	0.0860610	0.0430305	+	0.999074i	$0.486299\pi$
$920$	0	0
$921$	38.3535	1.26379
$922$	85.5985	2.81904
$923$	0	0
$924$	82.6486	2.71894
$925$	0	0
$926$	11.2459	0.369562
$927$	0.548487	0.0180147
$928$	11.7904	0.387040
$929$	11.1318	0.365222	0.182611	−	0.983185i	$-0.441545\pi$
$929$	11.1318	0.365222	0.182611	+	0.983185i	$0.441545\pi$
$930$	0	0
$931$	35.5033	1.16358
$932$	9.61557	0.314968
$933$	−16.1776	−0.529630
$934$	−63.6926	−2.08409
$935$	0	0
$936$	0	0
$937$	−51.5695	−1.68470	−0.842351	−	0.538930i	$-0.818829\pi$
$937$	−51.5695	−1.68470	−0.842351	+	0.538930i	$0.818829\pi$
$938$	44.1940	1.44298
$939$	−19.3916	−0.632822
$940$	0	0
$941$	−30.0426	−0.979362	−0.489681	−	0.871902i	$-0.662887\pi$
$941$	−30.0426	−0.979362	−0.489681	+	0.871902i	$0.662887\pi$
$942$	−35.7174	−1.16374
$943$	21.4015	0.696928
$944$	30.1239	0.980450
$945$	0	0
$946$	−47.7036	−1.55098
$947$	4.02504	0.130796	0.0653982	−	0.997859i	$-0.479168\pi$
$947$	4.02504	0.130796	0.0653982	+	0.997859i	$0.479168\pi$
$948$	32.3516	1.05073
$949$	0	0
$950$	0	0
$951$	−39.3852	−1.27715
$952$	−4.94245	−0.160186
$953$	−30.0216	−0.972496	−0.486248	−	0.873821i	$-0.661635\pi$
$953$	−30.0216	−0.972496	−0.486248	+	0.873821i	$0.661635\pi$
$954$	0.435589	0.0141027
$955$	0	0
$956$	−43.1905	−1.39688
$957$	12.3415	0.398944
$958$	−17.6202	−0.569284
$959$	28.2889	0.913497
$960$	0	0
$961$	−10.7455	−0.346630
$962$	0	0
$963$	−0.697144	−0.0224652
$964$	−36.0972	−1.16261
$965$	0	0
$966$	84.2068	2.70931
$967$	49.9590	1.60657	0.803286	−	0.595594i	$-0.203083\pi$
$967$	49.9590	1.60657	0.803286	+	0.595594i	$0.203083\pi$
$968$	14.2930	0.459394
$969$	7.16504	0.230174
$970$	0	0
$971$	−38.4043	−1.23245	−0.616227	−	0.787569i	$-0.711339\pi$
$971$	−38.4043	−1.23245	−0.616227	+	0.787569i	$0.711339\pi$
$972$	−2.15241	−0.0690384
$973$	−32.0709	−1.02814
$974$	43.4046	1.39077
$975$	0	0
$976$	−31.3715	−1.00418
$977$	23.0577	0.737680	0.368840	−	0.929493i	$-0.379755\pi$
$977$	23.0577	0.737680	0.368840	+	0.929493i	$0.379755\pi$
$978$	−73.0166	−2.33481
$979$	48.9611	1.56480
$980$	0	0
$981$	0.932796	0.0297819
$982$	−85.6953	−2.73465
$983$	−2.26226	−0.0721550	−0.0360775	−	0.999349i	$-0.511486\pi$
$983$	−2.26226	−0.0721550	−0.0360775	+	0.999349i	$0.511486\pi$
$984$	−9.16691	−0.292231
$985$	0	0
$986$	−3.00088	−0.0955675
$987$	29.6057	0.942358
$988$	0	0
$989$	−27.7088	−0.881088
$990$	0	0
$991$	−5.94809	−0.188947	−0.0944736	−	0.995527i	$-0.530117\pi$
$991$	−5.94809	−0.188947	−0.0944736	+	0.995527i	$0.530117\pi$
$992$	34.6984	1.10168
$993$	49.8557	1.58212
$994$	72.5078	2.29981
$995$	0	0
$996$	8.45070	0.267771
$997$	−25.4799	−0.806955	−0.403478	−	0.914989i	$-0.632199\pi$
$997$	−25.4799	−0.806955	−0.403478	+	0.914989i	$0.632199\pi$
$998$	−38.4079	−1.21578
$999$	−19.1277	−0.605175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.ca.1.17		18
5.2	odd	4		845.2.b.g.339.17	yes	18
5.3	odd	4		845.2.b.g.339.2	✓	18
5.4	even	2	inner	4225.2.a.ca.1.2		18
13.12	even	2		4225.2.a.cb.1.2		18
65.2	even	12		845.2.l.g.654.4		72
65.3	odd	12		845.2.n.i.529.17		36
65.7	even	12		845.2.l.g.699.33		72
65.8	even	4		845.2.d.e.844.34		36
65.12	odd	4		845.2.b.h.339.2	yes	18
65.17	odd	12		845.2.n.h.484.2		36
65.18	even	4		845.2.d.e.844.4		36
65.22	odd	12		845.2.n.i.484.17		36
65.23	odd	12		845.2.n.h.529.2		36
65.28	even	12		845.2.l.g.654.33		72
65.32	even	12		845.2.l.g.699.3		72
65.33	even	12		845.2.l.g.699.4		72
65.37	even	12		845.2.l.g.654.34		72
65.38	odd	4		845.2.b.h.339.17	yes	18
65.42	odd	12		845.2.n.i.529.2		36
65.43	odd	12		845.2.n.h.484.17		36
65.47	even	4		845.2.d.e.844.3		36
65.48	odd	12		845.2.n.i.484.2		36
65.57	even	4		845.2.d.e.844.33		36
65.58	even	12		845.2.l.g.699.34		72
65.62	odd	12		845.2.n.h.529.17		36
65.63	even	12		845.2.l.g.654.3		72
65.64	even	2		4225.2.a.cb.1.17		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.2	✓	18	5.3	odd	4
845.2.b.g.339.17	yes	18	5.2	odd	4
845.2.b.h.339.2	yes	18	65.12	odd	4
845.2.b.h.339.17	yes	18	65.38	odd	4
845.2.d.e.844.3		36	65.47	even	4
845.2.d.e.844.4		36	65.18	even	4
845.2.d.e.844.33		36	65.57	even	4
845.2.d.e.844.34		36	65.8	even	4
845.2.l.g.654.3		72	65.63	even	12
845.2.l.g.654.4		72	65.2	even	12
845.2.l.g.654.33		72	65.28	even	12
845.2.l.g.654.34		72	65.37	even	12
845.2.l.g.699.3		72	65.32	even	12
845.2.l.g.699.4		72	65.33	even	12
845.2.l.g.699.33		72	65.7	even	12
845.2.l.g.699.34		72	65.58	even	12
845.2.n.h.484.2		36	65.17	odd	12
845.2.n.h.484.17		36	65.43	odd	12
845.2.n.h.529.2		36	65.23	odd	12
845.2.n.h.529.17		36	65.62	odd	12
845.2.n.i.484.2		36	65.48	odd	12
845.2.n.i.484.17		36	65.22	odd	12
845.2.n.i.529.2		36	65.42	odd	12
845.2.n.i.529.17		36	65.3	odd	12
4225.2.a.ca.1.2		18	5.4	even	2	inner
4225.2.a.ca.1.17		18	1.1	even	1	trivial
4225.2.a.cb.1.2		18	13.12	even	2
4225.2.a.cb.1.17		18	65.64	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.15692 q^{2} +1.75445 q^{3} +2.65230 q^{4} +3.78422 q^{6} -3.86123 q^{7} +1.40696 q^{8} +0.0781086 q^{9} +O(q^{10})\)	\(q+2.15692 q^{2} +1.75445 q^{3} +2.65230 q^{4} +3.78422 q^{6} -3.86123 q^{7} +1.40696 q^{8} +0.0781086 q^{9} -4.59986 q^{11} +4.65334 q^{12} -8.32836 q^{14} -2.26990 q^{16} +0.909776 q^{17} +0.168474 q^{18} +4.48893 q^{19} -6.77435 q^{21} -9.92154 q^{22} -5.76296 q^{23} +2.46845 q^{24} -5.12632 q^{27} -10.2411 q^{28} -1.52926 q^{29} -4.50050 q^{31} -7.70991 q^{32} -8.07025 q^{33} +1.96231 q^{34} +0.207168 q^{36} +3.73128 q^{37} +9.68225 q^{38} -3.71363 q^{41} -14.6117 q^{42} +4.80809 q^{43} -12.2002 q^{44} -12.4302 q^{46} -4.37026 q^{47} -3.98243 q^{48} +7.90910 q^{49} +1.59616 q^{51} +2.58550 q^{53} -11.0571 q^{54} -5.43261 q^{56} +7.87561 q^{57} -3.29848 q^{58} -13.2710 q^{59} +13.8206 q^{61} -9.70721 q^{62} -0.301595 q^{63} -12.0899 q^{64} -17.4069 q^{66} -5.30644 q^{67} +2.41300 q^{68} -10.1108 q^{69} -8.70613 q^{71} +0.109896 q^{72} +11.4102 q^{73} +8.04806 q^{74} +11.9060 q^{76} +17.7611 q^{77} +6.95233 q^{79} -9.22822 q^{81} -8.01000 q^{82} +1.81605 q^{83} -17.9676 q^{84} +10.3707 q^{86} -2.68301 q^{87} -6.47184 q^{88} -10.6440 q^{89} -15.2851 q^{92} -7.89591 q^{93} -9.42629 q^{94} -13.5267 q^{96} +2.34480 q^{97} +17.0593 q^{98} -0.359289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9}+O(q^{10}) \) 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49} - 40 q^{51} - 90 q^{54} - 20 q^{56} - 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} - 72 q^{71} + 30 q^{74} - 4 q^{76} + 16 q^{79} - 30 q^{81} - 78 q^{84} + 30 q^{86} - 94 q^{89} - 128 q^{94} + 18 q^{96} - 76 q^{99}+O(q^{100}) \) 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9 - 22 * q^11 + 4 * q^14 - 12 * q^16 - 28 * q^19 - 26 * q^21 - 34 * q^24 - 20 * q^29 - 32 * q^31 - 18 * q^34 + 32 * q^36 - 52 * q^41 - 50 * q^44 - 30 * q^46 + 44 * q^49 - 40 * q^51 - 90 * q^54 - 20 * q^56 - 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 - 72 * q^71 + 30 * q^74 - 4 * q^76 + 16 * q^79 - 30 * q^81 - 78 * q^84 + 30 * q^86 - 94 * q^89 - 128 * q^94 + 18 * q^96 - 76 * q^99

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